Graph minors. XIII: the disjoint paths problem
01 Jan 1995-Journal of Combinatorial Theory, Series B (Academic Press, Inc.)-Vol. 63, Iss: 1, pp 65-110
TL;DR: An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
About: This article is published in Journal of Combinatorial Theory, Series B.The article was published on 1995-01-01 and is currently open access. It has received 1438 citations till now. The article focuses on the topics: Bound graph & Graph power.
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TL;DR: This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class.
1,197 citations
Journal Article•
TL;DR: A short overview of recent results in algorithmic graph theory that deal with the notions treewidth and pathwidth can be found in this paper, where the authors discuss algorithms that find tree-decomposition, algorithms that use treedecompositions to solve hard problems efficiently, graph minor theory, and some applications.
Abstract: A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography.
755 citations
01 Jun 1993
TL;DR: Every minor-closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm that determines whether the treewidth of G is at most k, and if so, finds a treedecomposition of G withtreewidth at mostK.
Abstract: In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.
727 citations
TL;DR: This work shows that INDEPENDENT SET is complete for W, and the W Hierarchy of parameterized problems was defined, and complete problems were identified for the classes W [ t ] for t ⩾ 2.
659 citations
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TL;DR: A polynomial-time algorithm to approximate the branch-width of certain symmetric sub-modular functions, and gives two applications to graph "clique-width" and the area of matroid branch- width.
577 citations
Cites background from "Graph minors. XIII: the disjoint pa..."
...• For fixed k there is a polynomial time algorithm that either decides that an input graph has tree-width at least k+1, or outputs a decomposition of tree-width at most 4k (this is an easy modification of the algorithm to estimate graph branchwidth presented in [20])....
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